Approximating geometrical graphs via spanners and banyans S. B. Rao - - PowerPoint PPT Presentation

Approximating geometrical graphs via “spanners” and “banyans”

• S. B. Rao & W. D. Smith

Proc of STOC, 1998: 540-550

Recap (Arora’s Algorithm)

Recap (Arora’s Algorithm)

Recap (Arora’s Algorithm)

Recap (Arora’s Algorithm)

Recap (Arora’s Algorithm)

Recap (Arora’s Algorithm)

Applications

Minimum Steiner Tree

• Shortest network connecting all sites
• Uses the same algorithm

Portals can be Steiner nodes! Min case different for Dynamic Programming Interface specification is slightly changed

k-TSP (The shortest tour that visits at least k nodes)

• Need to transform the instance
• Need assumption OPT>L (length of the grid)

Minimal Euclidean Matching

Motivations

Arora’s Algorithm,

Running time:

Several problems:

Monte Carlo succeeds with probability > ½. Need several runs. Derandomized version runs nd times slower.

Faster algorithm:

Las Vegas.

( )

1

/

( (log ) )

d

O d

O n n

ε

−

Results Summary

*Monte Carlo (d = 2)

• *Deterministic (d = 2)
• Monte Carlo (any d)
• Note: s = 1/

( ) (log )O s N N

( )

log

O s

s N sN N + ( ) 5(log )O s N N

(1)

(1)

2 log

O

O

s N s N N +

1

( ( ) )

( ) ( log )

d

O d ds

s d N O dN N

−

+

1

( ) (log )

d

O s d N N

−

Results Relevance

*Deterministic (d = 2)

Arora Rao & Smith

With s fixed, complexity is optimal for:

TST, MST, t-spanners, Min-Weight Matching.

Conjectured for Minimal Steiner Tree, Edge Cover, Nearest Neighbor With d fixed,

is likely optimal.

For large s comparible to N the approximation is exact and NP-Hard problems are assumed not soluble in time

( log ) O N N

( ) 5(log )O s N N

(1) (1)

2 log

O O s

N s N N +

(1)

(2 )

• N

O

(1)

2

O

s N

Rescale and Snap to grid

Assume the point set is in Assume the length of the Minimum Spanning Tree M 1 Scale by a factor L =

Key: Integer Coordinates in

Length of new MST Added a to the approximation factor

Algorithm (1)

/( ) d N M δ [0, ) , /

d

L L d N δ ≤ / d N δ ≤

[0,1]d δ

Algorithm (2)

Spanner

t-spanner: subgraph G’ of the complete Euclidean graph such that for any u, v, d(u,v) in G’ td(u,v) Claim 1: There is a -TST inside a -spanner

(1 ) ε +

(1 ) ε +

Algorithm (2)

Spanner

t-spanner: subgraph G’ of the complete Euclidean graph such that for any u, v, d(u,v) in G’ td(u,v) Claim 2: This TST does not use any edge more than twice

Algorithm (2)

Spanner

t-spanner: subgraph G’ of the complete Euclidean graph such that for any u, v, d(u,v) in G’ td(u,v) Claim 2: This TST does not use any edge more than twice

Replace each multiple edge > 2 by a multiple edge 2 with the same parity. This graph is still Eulerian and hence has a shorter Euler tour. Contradiction.

Algorithm (2)

Find a -spanner of the grid points

Has O (nsO(1)) edges Is sK longer than MST for some constant K Is guaranteed to contain a -TST Is computable in time*

(1 (1/ )) O s +

(1 (1/ )) O s +

(1)

( log )

O

O s N N

* S. Arya et al. Euclidean Spanners: short, think and lanky, Proc. TOC 1995

Algorithm (3)

Grow the grid by extending it randomly in each

direction by L

Subdivide the grid into

a quadtree

Algorithm (4)

Patch the Spanner with respect to the quadtree

Each quadtree square is intersected at most r times The total length of the added line segments is E(O(1/r))

Prop: If there was a path in the original spanner, there is a

path ' in the modified spanner that is longer by at most twice the increase of the cost of patching. (2O(1/r) total)

Algorithm (4)

Patch the Spanner with respect to the quadtree

Each quadtree square is intersected at most r times The total length of the added line segments is E(O(1/r)) Set r = O(sK+1). The added length is O(sKM)/sK+1 = O(M/s) Thus, if there existed a -TST in the original spanner, there must exist a -TST + 2 O(M/s)

• TST

(1 (1/ )) O s + (1 (1/ )) O s +

(1 (1/ )) O s +

Algorithm (5)

Find the shortest TST inside the modified spanner

with dynamic programming on the quadtree:

For each box of the quadtree: there are r points where the spanner crosses the boundary. at most 4 ways a tour can cross each point (enter/exit). 2O(r) matchup conditions on each side of the boundary

Thus, to get a solution in a larger box, consider all 2O(r) pairs

• f compatible boundary conditions in two smaller boxes

Algorithm Summary

Scale and snap the points Find a (1+)-spanner, = 1/(2s) Find a randomly shifted quad-tree Modify the spanner to make it r-light with

respect to the quadtree. Set r = cs4

Find the shortest r-light TST by dynamic

programming on the quadtree

(1)

2

O

s

N

N

log sN N log N sN

3

s N

(1)

(2 log )

O

s

O N sN N +

Total:

Algorithm Summary

Scale and snap the points Find a (1+)-spanner, = 1/(2s)

Total length

Modify the spanner to make it r-light

Choose r = cs4 With probability 1/2, the increase is bounded by 1/(4s) If so, output the graph, otherwise fail.

If the graph is produced, it is guaranteed to contain a

1+1/(2s)+2/(4s) = (1+1/s)-TST

3

( ( )/ ) O l MST ε

Derandomization

Unlike Arora, can average length increase over one dimension.

• Therefore, can optimize the quadtree shift for each dimension

independently

Arbitrarily fix one dimension. Dynamically, build a table of costs inflicted by possible shifts:

Create a table of costs for every line Aggregate costs for 2k lines by adding them

Modify the spanner, given the best shift. Repeat for all dimensions.

Intuitive Justification (Why can we do better?)

To prove (1+)-bound, Arora

• Needed to consider the expected change in the TSP path instead
• f the whole graph
• Harder bound
• Could only randomize the quadtree shift, or to search in all d

Rao & Smith:

• Create a (1+)-spanner of known length (

)

• Don’t need to rely on the points after the spanner is constructed

(r does not depend on N)

• Can average in one dimension (bound length increase in the

graph and not in the tour)

3

( ( ) / ) O l MST ε

Problems

Practicality

The bounds above are upper bounds. In-practice performance is not known (for d=2 faster practical algorithms exist)

• factor could be large even for d=2 and reasonable s

Could combine with heuristics (tour “cleaning up”, solving small subproblems via previous algorithms, etc.) Deterministic version could be interpreted as a local optimizer

Extendability

Not applicable to Minimum Matching (yet). How much can Steiner points help the spanner?

( )

( )

2

O d

sd

Thank You